What Does Tiling and Tessellation Mean in Math?

Tessellation is a technique used in all dimensions, but we’re going to look at tessellation in 2D—in a plane.

Theory

Tessellation in a Plane

Tessellation in a plane is tiling using regular polygons.

That means tessellation can be seen as a pattern that completely fills a plane, just like tiles fill a floor. This means that the pattern can be expanded indefinitely in all directions, without any holes in the floor. In order for there to be no holes, all angles that share a vertex must add up to $360$ °.

Not all polygons can be used for tessellation. Out of the regular polygons, only the triangle, the square and the hexagon can be put together so that the sum of the angles becomes $360$ °. For that reason, these are the only figures that can be used for tessellation.

Tessellation with hexagons

Example 1

If you have tiles that are shaped like regular pentagons, can you fill the planar space?

To fill the plane with polygons, the sum of all angles that share a vertex has to be exactly $360$ °. You need to find what the angles are at the corners of a regular pentagon. To do this, you use the interior angle formula, which gives you

v = 180 ° - \frac{360 °}{5} = 180 ° - 72 ° = 108 °

This means that the interior angles of a pentagon are $108$ °. If you put three pentagons next to each other, the sum of the angles is $324$ °, which is too small. If you put four pentagons next to each other, the sum of the angles is $432$ °, which is too big. For that reason, you can’t use regular pentagons to fill the plane.

Example 2

If you have regular triangles and hexagons, is there any way to put them together to fill the plane?

You begin by finding the angles at the corners of a regular triangle with interior angle formula. That gives you

v = 180 ° - \frac{360 °}{3} = 180 ° - 120 ° = 60 °

which means that the interior angles of a regular triangle are $60$ °.

Next, you find the interior angles of a regular hexagon with the same formula. You get

v = 180 ° - \frac{360 °}{6} = 180 ° - 60 ° = 120 °

which means that the interior angles of a regular hexagon are $120$ °.

If you put two triangles and two hexagons next to each other so that they meet at one point, you can see that the sum of the angles at that point is

120 ° + 120 ° + 60 ° + 60 ° = 360 °

That means you can use two regular triangles and two regular hexagons to fill the planar space.

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Examples of Geometric Patterns

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